Chapter I

Set Theory

There is surely a piece of divinity in us, something that was before the elements, and owes no homage unto the sun. Sir Thomas Browne

One of the benefits of mathematics comes from its ability to express a lot of information in very few symbols. Take a moment to consider the expression d sin( θ). dθ It encapsulates a large amount of information. The notation sin( θ) represents, for a right triangle with angle θ, the ratio of the opposite side to the hypotenuse. The differential operator d/dθ represents a limit, corresponding to a tangent line, and so forth. Similarly, sets are a convenient way to express a large amount of information. They give us a language we will find convenient in which to do mathematics. This is no accident, as much of modern mathematics can be expressed in terms of sets.

1 2 CHAPTER I. SET THEORY

1 Sets, subsets, and set operations

1.A What is a set? A set is simply a collection of objects. The objects in the set are called the elements . We often write down a set by listing its elements. For instance, the set S = 1, 2, 3 has three elements. Those elements are 1, 2, and 3. There is a special symbol,{ }, that we use to express the idea that an element belongs to a set. For instance, we∈ write 1 S to mean that “1 is an element of S.” For∈ the set S = 1, 2, 3 , we have 1 S, 2 S, and 3 S. We can write this more quickly as: 1 , 2{, 3 S}. We can express∈ the∈ fact that 4∈ is not an element of S by writing 4 / S. ∈ ∈ Example 1.1. Let P be the set 16 , 5, 2, 6, 9 . Is 6 P ? Yes! Is 5 P ? No, so we write 5 / P . { − } ∈ ∈ ∈ △ The order of the elements in a set does not matter, so we could have written S = 1, 2, 3 as S = 1, 3, 2 , or as S = 3, 2, 1 . If an element is repeated in a set, we do{ not count} the{ multiplicity.} Thus {1, 2, 3}, 1 is the same set as S = 1, 2, 3 . We say that two sets are equal when they{ have exactly} the same elements. { } Not all sets consist of numbers. For instance T = a, b, c, d is a set whose elements are the letters a, b, c, d . Sets may have words, names,{ symbols,} and even other sets as elements. Example 1.2. Suppose we want to form the set of Jesus’ original twelve apostles. This would be the set Apostles = Peter , James , John the beloved ,..., Judas Iscariot . { } We put the 3 dots in the middle to express the fact that there are more elements which we have not listed (perhaps to save time and space). △

The Last Supper , ca. 1520, by Giovanni Pietro Rizzoli.

The next example is a set with another set as an element. 1. SETS, SUBSETS, AND SET OPERATIONS 3

Example 1.3. Let S = 1, 5, 4, 6 , 3 . This set has four elements. We have 1, 5, 3, 4, 6 S, but 4 / S{. However,{ } 4 } 4, 6 and 4, 6 S. { } ∈ ∈ ∈ { } { } ∈ △ It can be confusing when sets are elements of other sets. You might ask why mathematicians would allow such confusion! It turns out that this is a very useful thing to allow; just like when moving, the moving truck (a big box) has boxes inside of it, each containing other things.

Advice 1.4. You can think of sets as boxes with objects inside. So we could view the set 1, 5, 4, 6 , 3 from the previous example as the following box, which contains{ another{ } box:}

1 5 4 6 3

Figure 1.4 : A box with a box inside, each containing some numbers.

1.B Naming sets We’ve seen that capitalized Roman letters can be used to give names to sets. Some sets are used so often that they are represented by special symbols. Here are a couple of examples. The set of natural numbers is the set • N = 1, 2, 3,... . { } This is the first example we’ve given of an infinite set , i.e., a set with infinitely many elements. The set of integers is • Z = ..., 3, 2, 1, 0, 1, 2, 3, 4,... . { − − − } The dots represent the fact that we are leaving elements unwritten in both directions. We use the fancy letter “ Z” because the word “integer” in German is “Zahlen.” Some sets are constructed using rules. For example, the set of even integers can be written as ..., 4, 2, 0, 2, 4,... { − − } but could also be written in the following ways:

(1.5) 2x : x Z { ∈ } (1.6) x Z : x is an even integer { ∈ } (1.7) x : x = 2 y for some y Z . { ∈ } 4 CHAPTER I. SET THEORY

We read the colon as “such that,” so ( 1.5 ) is read as “the set of elements of the form 2x such that x is an integer.” Writing sets with a colon is called set-builder notation . Notice that x Z : 2 x + 1 { ∈ } doesn’t make any sense, since “2 x + 1” is not a condition on x. Here are a few more examples. The set of prime numbers is

2, 3, 5, 7, 11 , 13 ,... = x N : x is prime . { } { ∈ } Similarly, Apostles = x : x was one of the original 12 apostles of Jesus . { } With set-builder notation, we can list a few more very important sets. The set of rational numbers is • Q = a/b : a, b Z, b = 0 . { ∈ 6 } Note that there is no problem with the fact that different fractions can represent the same rational number, such as 1 /2 = 2 /4. Repetitions do not matter in sets. We will occasionally need the fact that we can always write a rational number as a fraction a/b in lowest terms : i.e., so that a and b have no common factor larger than 1. We will prove this in Section 18 (see Exercise 18.3 ). The set of real numbers is • R = x : x has a decimal expansion . { } So we have π = 3 .14159 . . . R, 3 = 3 .00000 . . . R, and √2 R. Later in ∈ ∈ ∈ this book we will prove √2 / Q. The set of complex numbers ∈is • C = a + bi : a, b R, i 2 = 1 . { ∈ − } Is 3 a complex number? Yes, because we can take a = 3 and b = 0. So we have 3 N, 3 Z, 3 Q, 3 R, and 3 C! ∈ ∈ ∈ ∈ ∈ Example 1.8. Which of the named sets does π = 3 .14159 . . . belong to? We have π R and π C. On the other hand, since 3 < π < 4 we have π / N and π / Z. It ∈ ∈ ∈ ∈ is true, but much harder to show, that π / Q. ∈ △ There is one more set we will give a special name. The empty set is the set with no elements. We write it as = . • ∅ { }

Warning 1.9. The empty set is not nothing . It has no elements, but the empty set is something . Namely, it is “the set with nothing in it”. Thinking in terms of boxes, we can think of the empty set as an empty box. The box is something even if it has nothing in it. The symbol does not mean nothing. It means . ∅ { } 1. SETS, SUBSETS, AND SET OPERATIONS 5

Example 1.10. Sometimes we want the empty set to be an element of a set. For instance, we might take S = . {∅} The set S has a single element, namely . We could also write S = . In terms of boxes, S is the box containing an empty∅ box. Note that not all sets{{ have }} the empty set as an element. △

A box with an empty box inside, representing . {∅}

1.C Subsets In many activities in life we don’t focus on all the elements of a set, but rather on subcollections. To give just a few examples: The set of all phone numbers is too large for most of us to handle. The subcol- • lection of phone numbers of our personal contacts is much more manageable. If we formed the set of all books ever published in the world, this set would be • very large (but still finite!). However, the subcollection of books we have read is much smaller. If we want to count how many socks we own, we could use elements of the • integers Z, but since we cannot own a negative number of socks, a more natural set to use would be the subcollection of nonnegative integers

Z = 0, 1, 2, 3,... . ≥0 { } A subcollection of a set is called a subset . When A is a subset of B we write A B and if it is not a subset we write A * B. There are a couple different ways to think⊆ about the concept of A being a subset of B.

Option 1: To check that A is a subset of B, we check that every element of A also belongs to B. Example 1.11. (1) Let A = 1, 5, 6 and B = 1, 5, 6, 7, 8 . Is A a subset of B? Yes, because we can check{ that each} of A’s three{ elements,} 1, 5, and 6, belongs to B. (2) Let A = 6, 7/3, 9, π and B = 1, 2, 6, 7/3, π, 10 . Is A B? No, because { } { } ⊆ 9 A but 9 / B. So we write A * B. ∈ ∈ (3) Let A = N and B = Z. Is A B? Yes, every natural number is an integer. ⊆ (4) Is Z a subset of N? No, because Z has the element 1, which doesn’t belong − to N. △ 6 CHAPTER I. SET THEORY

Option 2: To check that A is a subset of B, we check that we can form A by throwing out some of the elements of B. Example 1.12. (1) Let A = 1, 5, 6 and B = 1, 5, 6, 7, 8 . Is A a subset of B? Yes, because we can throw{ away} 7 , 8 from B {to get A. } (2) Let A = 6, 7/3, 9, π and B = 1, 2, 6, 7/3, π, 10 . Is A a subset of B? No, because as{ we throw} away elements{ of B, we can never} get 9 inside. (3) Let A = N and B = Z. Is A a subset of B? Yes, because we can throw away the negative integers and 0 to get the natural numbers. (4) Is Z a subset of N? No, because we cannot get 1 by throwing away elements − from N. △ When we write A B, the little line segment at the bottom of “ ” means that there is possible equality.⊆ (Just like x y means that x is less than ⊆or equal to y.) Sometimes we do not want to allow equality.≤ We use the following terminology in this case.

Definition 1.13. If A S and A = S, we say that A is a proper subset of S, ⊆ 6 and we write A ( S.

Note that the symbol ( is different from *. If A ( B, then A is a subset of B that is not equal to B, while if A * B, then A is not a subset of B. Example 1.14. We have 1, 2 ( 1, 2, 3 . Of course 1, 2 1, 2, 3 is also true. { } { } { } ⊆ { } △

Warning 1.15. Some authors use instead of . Other authors use instead ⊂ ⊆ ⊂ of (. Thus, there can be a lot of confusion about what means, which is one reason why we will avoid that notation in this book! ⊂

Warning 1.16. Many students learning about subsets get confused about the difference between being an element and being a subset. Consider your music library as a set. The elements are the individual songs. Playlists, which are collections of some of the songs, are subsets of your library.

Example 1.17. The elements of C are complex numbers like 3+6 i or 2.7 5.9i. The − − subsets of C are sets of complex numbers like 5.4 7.3i, 9+0 i, 2.671+9 .359 i . { − − } △ Example 1.18. (1) Let T = 1, 2, 3, 4, 5 . Is 2 an element or a subset of T ? It is an element, since it lives{ inside T .} It is not a subset, since it isn’t a set of elements of T . (2) Let U = 5, 6, 7, 3 . Is 6 an element or a subset of U? It is not an element of U, since{− the set }6 isn’t{ } in its list of elements. It is a subset because it is a box whose elements{ come} from U. 1. SETS, SUBSETS, AND SET OPERATIONS 7

(3) Let X = 6 , 7, 8 , 5, 8 . Is 7 an element or a subset of X? Neither! It is not one of{{ the} three{ } elements{ }} listed in X, and it is not a box of elements in X either. Is 7, 8 an element or a subset of X? It is an element, since it is one of the three{ listed} elements. It is not a subset, even though it is a box, since it has elements which don’t belong to X. (4) Let Y = 5, 5 . Is 5 an element or a subset of Y ? It is both! It is an element, since{ { it}} is the{ second} element listed inside Y . It is also a subset of Y , since it is a box containing the first element of Y . △ It can be useful to construct sets satisfying certain properties in relation to one another. In the following example we show how this can be done. Example 1.19. We will find three sets A, B, C satisfying the following conditions: (1) A B, (2) A ⊆ C, and ∈ (3) C B with C = B (i.e., C ( B). One⊆ method to solve6 this problem is to start with the simplest sets possible and modify them as needed. So let’s start with A = , B = , C = . { } { } { } We see that condition (1) is fulfilled, but condition (2) is not. To force condition (2) to be true, we must make A an element of C. Thus, our new sets are A = , B = , C = A . { } { } { } Condition (1) still holds, and condition (2) is now true. However, condition (3) doesn’t hold. To make (3) true, we need B to have all the elements of C and at least one more. So we take A = , B = A, 1 , C = A . { } { } { } We double-check that all of the conditions hold (which they do), and so we have our final answer. △ 1.D Cardinality The number of elements of a set is called its cardinality . For instance, the set S = 1, 2, 3 has 3 elements. We write S = 3 to denote that S has cardinality 3. Note that{ } = 0 but = 1. A set| is |finite if its cardinality is either 0 or a natural number,|∅| and it is |{∅}|infinite otherwise. In a later section in the book, we will talk about a better way to define cardinality for infinite sets. Example 1.20. If T = 5, 6, 7, 8 , 3 , 0, , the cardinality is T = 5. { { } { } ∅} | | △ In mathematics, we sometimes use the same symbols for two different things. The meaning of the symbols must be deduced from their context. For instance, if we write 3.392 this is certainly not the cardinality of a set, but instead is probably referring |−to the absolute| value of a number. In the next example, we use in two different ways. | · | 8 CHAPTER I. SET THEORY

Example 1.21. If T = x Z : x < 4 , what is T ? (Hint: It is bigger than 4.){ ∈ | | } | | △ 1.E Power sets In this section we define the power set and give some examples.

Definition 1.22. Let S be a set. The power set of S is the new set P(S) whose elements are the subsets of S. In other words, A P(S) exactly when A S. ∈ ⊆

The next example determines the power set of a small set S. Example 1.23. Can we list all of the subsets of S = 1, 2, 3 ? If we think about subsets as “boxes containing only elements of S”, we just{ have to} list all possibilities. They are as follows:

P(S) = , 1 , 2 , 3 , 1, 2 , 1, 3 , 2, 3 , 1, 2, 3 . {∅ { } { } { } { } { } { } { }} Why is the empty set one of the subsets? Is it really a box containing only elements of S? Yes, its elements (there are none!) all belong to S. Thinking about it in terms of “throwing away” elements of S, we threw all of them away. Why is S S? Because S is a box containing only elements of S. Thinking in terms of “throwing⊆ away” elements, we threw away none of the elements. △ If S is a finite set, we can determine the size of the power set P(S) from S . | | | |

Theorem 1.24. If S = n, then P(S) = 2 n. | | | |

Here is a sketch of why this is true. To form a subset of S, for each element in S we choose to keep or throw away that element. Thus, there are 2 choices for each element. Since there are n elements, this gives 2 n options. Example 1.25. For the set S = 1, 2, 3 we have S = 3. Thus the power set has cardinality P(S) = 2 3 = 8. This{ is exactly} the| number| of elements we listed in Example 1.23| . | △ Example 1.26. How may elements will the power set of U = 1, have? The set U has two elements, so there should be 2 2 = 4 subsets. They can{ be∅} listed as:

P(U) = , 1 , , U . {∅ { } {∅} } Which of these are proper? (All of them except U itself.) △ Example 1.27. List three elements of P(N), each having different cardinality, and one being infinite. Here is one possible answer: 1, 7 , 67 , 193 , 91948 , and 2, 4, 6, 8, 10 ,... . There are many other correct choices. { } { } { } △ 1. SETS, SUBSETS, AND SET OPERATIONS 9

1.F Unions and intersections There are multiple ways to modify sets. When there are two sets S and T , we can put them together to form a new set called the union , and we write

S T = x : x S or x T . ∪ { ∈ ∈ } This is the set of elements which belong to S or T or both of them. (When we use the word “or” in this book, we will almost always use the inclusive meaning.) Pictorially, we can view this set using a Venn diagram as follows.

S T

Figure 1.28: The union of S and T .

Similarly, given two sets S and T we can form the set of elements that belong to both of them, called the intersection , and we write

S T = x : x S and x T . ∩ { ∈ ∈ } The Venn diagram is the following.

S T

Figure 1.29: The intersection of S and T .

Example 1.30. Let A = 1, 6, 17 , 35 and B = 1, 5, 11 , 17 . Then { } { } A B = 1, 5, 6, 11 , 17 , 35 , A B = 1, 17 . ∪ { } ∩ { } △ Example 1.31. Find sets P, Q with P = 7, Q = 9, and P Q = 5. How big is P Q ? | | | | | ∩ | | ∪ | 10 CHAPTER I. SET THEORY

We start by letting P be the easiest possible set with 7 elements, namely P = 1, 2, 3, 4, 5, 6, 7 . Since Q must share 5 of these elements, but have 9 elements total, we{ could write Q} = 1, 2, 3, 4, 5, 8, 9, 10 , 11 . { } For the example we constructed, we have P Q = x N : x 11 , so P Q = 11. If we chose other sets P and Q, could ∪P Q be{ different?∈ (Answer:≤ } No. |The∪ given| numbers determine the cardinality of each| piece∪ | in the Venn diagram.) △

1.G Complements and differences Let S and T be sets. The difference of T and S is

T S = x : x T and x / S . − { ∈ ∈ } The Venn diagram is as follows.

S T

Figure 1.32: The difference of T and S.

Example 1.33. Let S = 1, 2, 3, 4, 5, 6, 7 and T = 6, 7, 8, 9 . We find T S is the set { } { } − T S = 8, 9 . − { } Notice that we do not need to worry about those elements of S which do not belong to T . We only have to take away the part the two sets share. So T S = T (S T ). − − ∩ Also notice that S T = 1, 2, 3, 4, 5 is different from T S. − { } − △ Example 1.34. Let A and B be sets. Assume A = 16 and B = 9. If A B = 2, what are A B and B A ? | | | | | ∩ | | − | | − | There are only two elements that A and B share, thus A B = 14 and B A = 7. | − | | − | Can you now figure out A B ? (Hint: Draw the Venn diagram.) | ∪ | △ Occasionally we will be working inside some set U, which we think of as the universal set for the problem at hand. For instance, when solving quadratic equations, such as x2 x + 2 = 0, your universal set might be the complex numbers C. − Given a subset S of the universal set U, we write S = U S, and call this the complement of S (in the universal set U). The Venn diagram follows.− 1. SETS, SUBSETS, AND SET OPERATIONS 11

U

S

Figure 1.35: The complement of a set S inside a universal set U.

Example 1.36. Let U = N and let P = 2, 3, 5, 7,... be the set of primes. What is P ? This is the set of composite numbers{ and 1,} or in other words P = 1, 4, 6, 8, 9,... . { } △ 1.H Exercises Exercise 1.1. Each of the following sets is written in set-builder notation. Write the set by listing its elements. Also state the cardinality of each set. (a) S = n N : 5 < n < 11 . 1 { ∈ | | } (b) S = n Z : 5 < n < 11 . 2 { ∈ | | } (c) S = x R : x2 + 2 = 0 . 3 { ∈ } (d) S = x C : x2 + 2 = 0 . 4 { ∈ } (e) S = t Z : t5 < 1000 . (This one is slightly tricky.) 5 { ∈ } Exercise 1.2. Rewrite each of the following sets in the form x S : some property on x , just as we did in ( 1.6 ) above,{ ∈ by finding an appropriate property.} (a) A = 1, 3, 5, 7, 9,... where S = N. 1 { } (b) A2 = 1, 8, 27 , 64 ,... where S = N. (c) A = { 1, 0 where }S = 1, 0, 1 . 3 {− } {− } Exercise 1.3. Write the following sets in set-builder notation. (a) A = ..., 10 , 5, 0, 5, 10 , 15 ,... . (b) B = {..., −7, −2, 3, 8, 13 , 18 ,... }. (c) C = {1, 16 −, 81 ,−256 ,... . } (d) D = {..., 1/4, 1/2, 1, 2}, 4, 8, 16 ,... . { } Exercise 1.4. Give specific examples of sets A, B, and C satisfying the following conditions (in each part, separately): (a) A B, B C, and A / C. ∈ ∈ ∈ (b) A B, B C, and A * C. ∈ ⊆ (c) A ( B, B C, and A C. ∈ ∈ (d) A B C, A * C, and B * C. (e) A ∩ C =⊆ , A B, B C = 3. ∩ ∅ ⊆ | ∩ | 12 CHAPTER I. SET THEORY

Exercise 1.5. Let A = 1, 2 . Find P(A), and then find P(P(A)). What are the cardinalities of these three{ sets?}

Exercise 1.6. Let a, b R with a < b . The closed interval [ a, b ] is the set x R : ∈ { ∈ a x b . Similarly, the open interval ( a, b ) is the set x R : a

A

B C

For each of the following sets, copy the Venn diagram above, and then shade in the named region: (a) A (B C). (b) A − (B ∩ C). (c) B − (A − C). (d) ( B− C)− (B A). (e) ( A ∩ B) ∩ (A ∪ C). − ∪ − Exercise 1.8. Two sets S, T are disjoint if they share no elements. In other words S T = . Which of the following sets are disjoint? Give reasons. (a)∩ The∅ set of odd integers and the set of even integers. (b) The natural numbers and the complex numbers. (c) The prime numbers and the composite numbers. (d) The rational numbers and the irrational numbers (i.e., real numbers which are not rational).

Exercise 1.9. Find some universal set U and subsets S, T U, such that S T = 3, T S = 1, S T = 6, and S = 2. (Write each of U⊆, S, and T by listing| − | their elements.)| − | | ∪ | | | 2. PRODUCTS OF SETS AND INDEXED SETS 13

2 Products of sets and indexed sets

2.A Cartesian products Sets are unordered lists of elements. There are situations where order matters. For instance, you probably don’t want to put your shoes on before your socks. To give a more mathematical example, if we square a number and then take its cosine, that is not the same as first taking the cosine and then squaring:

cos( x2) = (cos( x)) 2. 6 There are other situations where we want to keep things ordered. We will write ( x, y ) for the ordered pair where x occurs first and y occurs second. Thus ( x, y ) = ( y, x ) even though x, y = y, x . Also, an element can be repeated in an ordered6 list, such as (1 , 1),{ while} sets{ do} not count repetitions. There is a very nice notation for sets of ordered pairs.

Definition 2.1. Let S and T be two sets. The Cartesian product of these sets is the new set S T = (s, t ) : s S, t T . × { ∈ ∈ } This is the set of all ordered pairs such that the first entry comes from S and the second entry comes from T . We will often refer to S T just as the product of S and T . ×

We will now give an example of how to find simple Cartesian products.

Example 2.2. Let S = 1, 2, 3 and T = 1, 2 . What is S T ? It is the set (1 , 1) , (1 , 2) , (2 , 1) , (2 , 2) , (3{, 1) , (3}, 2) . Notice{ that} 3 can occur as× a first coordinate since{ 3 S, but not as a second coordinate} since 3 / T . ∈ ∈ While the order matters inside an ordered pair, we could have listed the elements of S T in a different order since S T is itself just a set (and order is irrelevant in sets).× So we could have written ×

S T = (1 , 2) , (2 , 2) , (3 , 1) , (1 , 1) , (2 , 1) , (3 , 2) . × { } However, S T = T S = (1 , 1) , (1 , 2) , (1 , 3) , (2 , 1) , (2 , 2) , (2 , 3) . × 6 × { } △ You might notice that in the previous example we have S T = 6 = 3 2 = S T . This is not an accident. In fact, the following is true, altho| ugh× we| do not· as yet| |·| have| the tools to prove it.

Proposition 2.3. Let A and B be finite sets, with A = m and B = n. Then A B is a finite set, with A B = mn . | | | | × | × | 14 CHAPTER I. SET THEORY

Sets do not need to be finite in order to act as components in products.

Example 2.4. Let A = N and B = 0, 1 . What are the elements of A B? They are { } ×

A B = (1 , 0) , (1 , 1) , (2 , 0) , (2 , 1) , (3 , 0) , (3 , 1) ,... = (n, 0) , (n, 1) : n N . × { } { ∈ } Is A B the same set as B A? No, they have different elements. For instance, (1 , 0) × A B, but (1 , 0) / B× A since 0 / A. ∈ × ∈ × ∈ △ In each of the previous examples, we took the Cartesian product of two different sets. If we take the product of a set with itself, we sometimes write A2 = A A. The following example is one of the most useful products of a set with itself. ×

Example 2.5. The set R2 = R R is called the Cartesian plane . We view elements × in this set as points (x, y ) : x, y R . { ∈ } y

x

The Cartesian plane, R R ×

The set S T in Example 2.2 is a subset of R R. We can graph it as follows: × × y

x

1, 2, 3 1, 2 { } × { } 2. PRODUCTS OF SETS AND INDEXED SETS 15

Similarly, the set A B from Example 2.4 is graphed as: × y

· · ·

x · · ·

N 0, 1 × { } We can now describe more complicated sets. For instance

(x, y ) R R : y = 3 x + 1 { ∈ × } is a line. The set R2 (0 , 0) is the punctured plane (the plane with the origin removed). Can you describe− { a simple} parabola? △ The following example addresses the question: “What do we do if one of the sets has no elements?” Example 2.6. We determine 1, 2, 3 . Elements of this set are ordered pairs of the form ( a, b ), with a 1, 2,{3 and }b × ∅ . Thus, there are no possible choices for b, and so 1, 2, 3 =∈ { . Note} that 3 ∈0 ∅= 0, so Proposition 2.3 works in this case too. { } × ∅ ∅ · △ Just as with ordered pairs, we can form the set of ordered triples A B C = (a, b, c ) : a A, b B, c C . × × { ∈ ∈ ∈ } We can similarly form ordered quadruples, ordered quintuples, and so forth. The next subsection will give us the tools necessary to talk about even more complicated constructions.

2.B Indices When we have a large number of sets, rather than writing them using different letters of the alphabet A,B,C,D,...,Z it can be easier to use subscripts

A1, A 2, A 3, A 4, . . . , A 26 . This notation is extremely powerful for the following reasons: 16 CHAPTER I. SET THEORY

The notation tells us how many sets we are working with, using a small number • of symbols. For instance, if we write A1, A 2, . . . , A 132 , we know that there are exactly 132 sets. (Try writing them down using different letters of the alphabet!) We can even talk about an infinite number of sets A , A , A ,... . Notice that • 1 2 3 the subscripts all come from the set N. We refer to N as the index set for this collection. Using indices we can form complicated unions, intersections, and Cartesian • products. Example 2.7. Let A = 1, 2, 4 , A = 3, 1, 5, 9 , and A = 1, 6, 10 . We find 1 { } 2 {− } 3 { } 3 [ Ai = A1 A2 A3 = 3, 1, 2, 4, 5, 6, 9, 10 ∪ ∪ {− } i=1 and 3 \ Ai = A1 A2 A3 = 1 . ∩ ∩ { } i=1 Those who have seen summation notation 3 2 2 2 2 X i = 1 + 2 + 3 i=1 will recognize where this notation comes from. △ Can we form infinite unions and intersections? This is actually a common occur- rence. Example 2.8. Let B = 1, 1 , B = 2, 2 , B = 3, 3 , and so forth. In other 1 { − } 2 { − } 3 { − } words Bn = n, n for each n N. (Notice that while the subscripts come from N, { − } ∈ the elements of the sets Bn come from Z.) The union is the set of elements which belong to at least one of the sets, thus ∞ [ Bn = ..., 3, 2, 1, 1, 2, 3,... = Z 0 . { − − − } − { } n=1 The intersection is the set of elements which belong to every one of the sets, thus ∞ \ Bn = = . { } ∅ △ n=1 There is an alternate way to write intersections and unions, using index sets. For instance, using the notation in the previous two examples, we could also write 3 \ Ai = A1 A2 A3 = \ Ai ∩ ∩ i=1 i∈{ 1,2,3} and ∞ [ Bn = [ Bn. n=1 n∈N There is nothing to limit our index set, so we can make the following broad definition. 2. PRODUCTS OF SETS AND INDEXED SETS 17

Definition 2.9. Let I be any set, and let Si be a set for each i I. We put ∈

[ Si = x : x belongs to Si for some i I { ∈ } i∈I

and

\ Si = x : x belongs to Si for each i I . { ∈ } i∈I

The next example shows, once again, how mathematics has the uncanny ability to express information in varied subjects using very simple notation. Example 2.10. Let A = a,b,c,d ,..., z be the “lowercase English alphabet set.” This set has twenty-six elements.{ Let V }= a,e,i,o,u be the “standard vowel set.” { } Notice that V ( A. Given α A, we let Wα be the set of words in the English language containing the letter α.∈ Note that α is a dummy variable , standing in for an actual element of A. For instance, if α = x then we have W = xylophone , existence , axiom ,... , x { } while if α = t then we have W = terminator , atom , attribute ,... . t { } Each set of words Wα is a subset of the universal set of all words in the English language. Try to answer the following questions: (1) What is Tα∈V Wα? (2) Is that set empty? (3) What is Sα∈V Wα? (4) Is that set empty? Here are the answers. (Look at them only after you have your own!) (1) This is the set of words that contain every standard vowel. (2) It isn’t empty, since it contains words like “sequoia,” “evacuation,” etc. (3) This is the set of words with no standard vowels. (Don’t forget that there is a bar over the union.) (4) It isn’t empty, since it contains words like “why,” “tsktsk,” etc. △ We finish with one more difficult example. Example 2.11. We determine Sx∈[1 ,2] [x, 2] [3 , x + 3]. First, to get a footing on this problem,× we try to understand what happens for certain values of x. The smallest possible x value in the union is when x = 1. There we get that [ x, 2] [3 , x + 3] = [1 , 2] [3 , 4]. This is the set of ordered pairs × × (x, y ) R2 : 1 x 2, 3 y 4 . This is just a box in the plane. Its graph is {the first∈ graph below.≤ ≤ ≤ ≤ } 18 CHAPTER I. SET THEORY

y y

x x

[1 , 2] [3 , 4] 2 [3 , 5] × { } ×

The largest possible x value in the union is when x = 2. There we get that [x, 2] [3 , x + 3] = [2 , 2] [3 , 5]. Notice that [2 , 2] = 2 is just a single point. Now 2 ×[3 , 5] is a line segment× in the plane, where the { x}-value is 2 and the y-values {range} × from 3 to 5. Its graph is the second graph above. If we consider the intermediate value x = 1 .5, we get the box [1 .5, 2] [3 , 4.5], graphed below on the left. ×

y y

x x

[1 .5, 2] [3 , 4.5] [ ([ x, 2] [3 , x + 3]) × × x∈[1 ,2]

Taking the union over all x [1 , 2], we get the region graphed above on the right, boxed in by the lines x = 1, y =∈ 3, x = 2, and y = x + 3. △ 2.C Exercises

Exercise 2.1. Sketch each of the following sets in the Cartesian plane R2. (a) 1, 2 1, 3 . (b){ [1 , 2] } ×[1 { , 3]. } (c) (1 , 2] × [1 , 3]. (Hint: If an edge is missing, use a dashed, rather than solid, line for that× edge.) (d) (1 , 2] 1, 3 . × { } 2. PRODUCTS OF SETS AND INDEXED SETS 19

Exercise 2.2. Let A = s, t and B = 0, 9, 7 . Write the following sets by listing all of their elements. { } { } (a) A B. (b) B × A. (c) A2×. (d) B2. (e) A. ∅ × Exercise 2.3. Answer each of the following questions with “True” or “False” and then provide a reason for your answer. (a) If A = 3 and B = 4, then A B = 7. (b) It| is| always true| that| A B |= B× |A when A and B are sets. × × (c) Assume I is an indexing set, and let Si be a set for each i I. We always have ∈ Si Si. Ti∈I ⊆ Si∈I (d) There exist distinct sets S1, S 2, S 3,... , each of which is infinite, but

∞ \ Si i=1 has exactly one element. (e) The set A4 consists of ordered triples from A.

Exercise 2.4. Using the notations from Example 2.10 , write the following sets (pos- sibly using intersections or unions). (a) The set of words containing all four of the letters “a,w,x,y.” (b) The set of words not containing any of the letters “s,t,u.” (c) The set of words containing both “p,r” but not containing any of the standard vowels. (Is this set empty?)

Exercise 2.5. For each number r R, consider the “parabola shifted by r” defined as: ∈ 2 2 Pr = (x, y ) R : y = x + r . { ∈ } Describe the following sets in set-builder notation; the answer should have no reference to “ r.” Also graph the sets in the Cartesian plane. (a) Sr∈R Pr. (b) Sr> 0 Pr. (c) Sr6=0 Pr. (d) Tr∈R Pr. (e) Tr> 0 Pr. 20 CHAPTER I. SET THEORY